That's The Very Definition?
True or False?
If f ( x ) is an odd function, then ∫ − ∞ ∞ f ( x ) d x = 0 .
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1 solution
Not entirely correct. If f ( x ) = x , then the statement is false as well.
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As it is with f ( x ) = sin ( x ) . For the problem statement to be true the function must be integrable, so besides f ( x ) being odd, it must at least be the case that x → ∞ lim f ( x ) = 0 . It may be possible to have a discontinuity, (say at x = 0 ); I'm trying to think of an example .....
Oh. I see why now. This is also probably due to the divergence of the improper integral as well, right?
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No, no. You're missing the point here. Do you know why ∫ − ∞ ∞ x d x diverges and why z → ∞ lim ∫ − z z x d x = 0 ?
Provided if the function is continuous everywhere and is odd, the definition is correct.
If the function is not continuous everywhere, say, for example, tan (x), which contains periodic infinite discontinuities every 2π period from +/- π/2, the definition would be false.